These tables give Poisson outcomes for selected
cases where the rate r (conventionally
called lambda, l) lies
between 0.1 and 5.0. For other values of r,
see the standard references (or our secret Forbidden
Poisson table). Empty spaces in the table mean that the probability
of that number of occurrences is effectively zero, and does not show up
within the limits of four decimal places.

The likeliest number
of occurrences (which is r, rounded down to
the nearest whole number) is shown in bold.
It is standard, for whole numberr,
that p(r) = p(r-1).
Notice that in the first of the two tables, with r
< 1, the likeliest occurrence per module of observation is 0. This is therefore
the range that most fully meets the Poisson criterion of "rare event."

Table
for r = 0.1 ~ 0.9

r
= 0.1

r
= 0.2

r
= 0.3

r
= 0.4

r
= 0.5

r
= 0.6

r
= 0.7

r
= 0.8

r
= 0.9

p(0)

0.9048

0.8187

0.7408

0.6703

0.6065

0.5488

0.4966

0.4493

0.4066

p(1)

0.0905

0.1637

0.2222

0.2681

0.3033

0.3293

0.3476

0.3595

0.3659

p(2)

0.0045

0.0164

0.0333

0.0536

0.0758

0.0988

0.1217

0.1438

0.1647

p(3)

0.0002

0.0011

0.0033

0.0072

0.0126

0.0198

0.0284

0.0383

0.0494

p(4)

0.0001

0.0003

0.0007

0.0016

0.0030

0.0050

0.0077

0.0111

p(5)

0.0001

0.0002

0.0004

0.0007

0.0012

0.0020

p(6)

0.0001

0.0002

0.0003

Table
for r = 1.0 ~ 5.0

Note that values higher
than r = 5 may be theoretically dubious:
5.0 is the highest value of r for which
the chance of getting zero occurrences per unit of observation is even remotely
likely: p ~ 0.01. If zero occurrences are unlikely to occur, then the event
cannot validly be called "rare."

r
= 1.0

r
= 1.5

r
= 2.0

r
= 2.5

r
= 3.0

r
= 3.5

r
= 4.0

r
= 4.5

r
= 5.0

p(0)

0.3679

0.2231

0.1353

0.0821

0.0498

0.0302

0.0183

0.0111

0.0067

p(1)

0.3679

0.3347

0.2707

0.2052

0.1494

0.1057

0.0733

0.0500

0.0337

p(2)

0.1839

0.2510

0.2707

0.2565

0.2240

0.1850

0.1465

0.1125

0.0842

p(3)

0.0613

0.1255

0.1804

0.2138

0.2240

0.2158

0.1954

0.1687

0.1404

p(4)

0.0153

0.0471

0.0902

0.1336

0.1680

0.1888

0.1954

0.1898

0.1755

p(5)

0.0031

0.0141

0.0361

0.0668

0.1008

0.1322

0.1563

0.1708

0.1755

p(6)

0.0005

0.0035

0.0120

0.0278

0.0504

0.0771

0.1042

0.1281

0.1462

p(7)

0.0001

0.0008

0.0034

0.0099

0.0216

0.0385

0.0595

0.0824

0.1044

p(8)

0.0001

0.0009

0.0031

0.0081

0.0169

0.0298

0.0463

0.0653

p(9)

0.0002

0.0009

0.0027

0.0066

0.0132

0.0232

0.0363

p(10)

0.0002

0.0008

0.0023

0.0053

0.0104

0.0181

p(11)

0.0002

0.0007

0.0019

0.0043

0.0082

p(12)

0.0001

0.0002

0.0006

0.0016

0.0034

p(13)

0.0001

0.0002

0.0006

0.0013

p(14)

0.0001

0.0002

0.0005

p(15)

0.0001

0.0002

Notice the skew quality visible
in both tables: the range of variation upward from r is greater than the
variation downward from r. The possible
values of r are constrained by having zero
on the downward side.

Conventional Poisson tables do not go beyond
r = 20,
this being the point at which the Poisson Distribution becames nearly indistinguishable
from the Normal Distribution. For practical purposes, then, the distinctive
qualities of the Poisson are more or less lost at that point. It has moved
too far out from its initial closeness to the "zero" barrier,
the point below which further variation cannot occur. There is too much
room on both sides of the most likely occurrence number, and the resulting
curve becomes more and more symmetrical.